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-rw-r--r--buch/papers/zeta/presentation/presentation.tex368
-rw-r--r--buch/papers/zeta/presentation/youtube_screenshot.pngbin0 -> 378662 bytes
-rw-r--r--buch/papers/zeta/presentation/zeta_color_plot-img0.pngbin0 -> 37362 bytes
-rw-r--r--buch/papers/zeta/presentation/zeta_color_plot.pgf402
-rw-r--r--buch/papers/zeta/primzahlfunktion.pgf505
-rw-r--r--buch/papers/zeta/python/plot_zeta.py39
-rw-r--r--buch/papers/zeta/python/plot_zeta2.py31
-rw-r--r--buch/papers/zeta/python/primzahlfunktion.py24
-rw-r--r--buch/papers/zeta/zeta_re_-1_plot.pgf1147
-rw-r--r--buch/papers/zeta/zeta_re_0.5_plot.pgf1206
-rw-r--r--buch/papers/zeta/zeta_re_0_plot.pgf1242
11 files changed, 4964 insertions, 0 deletions
diff --git a/buch/papers/zeta/presentation/presentation.tex b/buch/papers/zeta/presentation/presentation.tex
new file mode 100644
index 0000000..e106089
--- /dev/null
+++ b/buch/papers/zeta/presentation/presentation.tex
@@ -0,0 +1,368 @@
+\documentclass[ngerman, aspectratio=169]{beamer}
+
+%style
+\mode<presentation>{
+ \usetheme{Frankfurt}
+}
+%packages
+\usepackage[utf8]{inputenc}
+\usepackage[english]{babel}
+\usepackage{graphicx}
+\usepackage{array}
+
+\newcolumntype{L}[1]{>{\raggedright\let\newline\\\arraybackslash\hspace{0pt}}m{#1}}
+\usepackage{ragged2e}
+
+\usepackage{bm} % bold math
+\usepackage{amsfonts}
+\usepackage{amssymb}
+\usepackage{mathtools}
+\usepackage{amsmath}
+\usepackage{multirow} % multi row in tables
+\usepackage{scrextend}
+
+\usepackage{tikz}
+
+\usepackage{algorithmic}
+
+%\usepackage{algorithm} % http://ctan.org/pkg/algorithm
+%\usepackage{algpseudocode} % http://ctan.org/pkg/algorithmicx
+
+%\usepackage{algorithmicx}
+
+
+%citations
+\usepackage[style=verbose,backend=biber]{biblatex}
+\addbibresource{references.bib}
+
+
+
+\usefonttheme[onlymath]{serif}
+
+%Beamer Template modifications
+%\definecolor{mainColor}{HTML}{0065A3} % HSR blue
+\definecolor{mainColor}{HTML}{D72864} % OST pink
+\definecolor{invColor}{HTML}{28d79b} % OST pink
+\definecolor{dgreen}{HTML}{38ad36} % Dark green
+
+%\definecolor{mainColor}{HTML}{000000} % HSR blue
+\setbeamercolor{palette primary}{bg=white,fg=mainColor}
+\setbeamercolor{palette secondary}{bg=orange,fg=mainColor}
+\setbeamercolor{palette tertiary}{bg=yellow,fg=red}
+\setbeamercolor{palette quaternary}{bg=mainColor,fg=white} %bg = Top bar, fg = active top bar topic
+\setbeamercolor{structure}{fg=black} % itemize, enumerate, etc (bullet points)
+\setbeamercolor{section in toc}{fg=black} % TOC sections
+\setbeamertemplate{section in toc}[sections numbered]
+\setbeamertemplate{subsection in toc}{%
+ \hspace{1.2em}{$\bullet$}~\inserttocsubsection\par}
+
+\setbeamertemplate{itemize items}[circle]
+\setbeamertemplate{description item}[circle]
+\setbeamertemplate{title page}[default][colsep=-4bp,rounded=true]
+\beamertemplatenavigationsymbolsempty
+
+\setbeamercolor{footline}{fg=gray}
+\setbeamertemplate{footline}{%
+ \hfill\usebeamertemplate***{navigation symbols}
+ \hspace{0.5cm}
+ \insertframenumber{}\hspace{0.2cm}\vspace{0.2cm}
+}
+
+\usepackage{caption}
+\captionsetup{labelformat=empty}
+
+%Title Page
+\title{Riemannsche Zeta Funktion}
+\author{Raphael Unterer}
+\institute{Mathematisches Seminar 2022: Spezielle Funktionen}
+
+\newcommand*{\HL}{\textcolor{mainColor}}
+\newcommand*{\RD}{\textcolor{red}}
+\newcommand*{\BL}{\textcolor{blue}}
+\newcommand*{\GN}{\textcolor{dgreen}}
+\newcommand*{\YE}{\textcolor{violet}}
+
+
+
+
+\makeatletter
+\newcount\my@repeat@count
+\newcommand{\myrepeat}[2]{%
+ \begingroup
+ \my@repeat@count=\z@
+ \@whilenum\my@repeat@count<#1\do{#2\advance\my@repeat@count\@ne}%
+ \endgroup
+}
+\makeatother
+
+
+
+
+\usetikzlibrary{automata,arrows,positioning,calc}
+
+
+\begin{document}
+
+ %Titelseite
+ \begin{frame}
+ \titlepage
+ \end{frame}
+
+ %Inhaltsverzeichnis
+% \begin{frame}
+% \frametitle{Inhalt}
+% \tableofcontents
+% \end{frame}
+
+ \section{Motivation}
+
+ \begin{frame}
+ \frametitle{Summe aller Natürlichen Zahlen}
+ \begin{equation*}
+ \sum_{n=1}^{\infty} n
+ =
+ 1 + 2 + 3 + \ldots + \infty
+ =
+ - \frac{1}{12}
+ \end{equation*}
+ \end{frame}
+ \begin{frame}
+ \frametitle{Summe aller Natürlichen Zahlen}
+ \begin{center}
+ \includegraphics[width=0.7\textwidth]{youtube_screenshot.png}
+ \end{center}
+ \end{frame}
+ \begin{frame}
+ \frametitle{Riemannsche Zeta Funktion}
+ \begin{equation*}
+ \zeta(s)
+ =
+ \sum_{n=1}^{\infty}
+ \frac{1}{n^s}
+ \end{equation*}
+ \pause
+ \begin{equation*}
+ \zeta(-1)
+ =
+ \sum_{n=1}^{\infty}
+ \frac{1}{n^{-1}}
+ =
+ \sum_{n=1}^{\infty} n
+ \end{equation*}
+ \end{frame}
+ \begin{frame}
+ \frametitle{Originaler Definitionsbereich}
+ Wir kennen die divergierende harmonische Reihe
+ \begin{equation*}
+ \zeta(1)
+ =
+ \sum_{n=1}^{\infty}
+ \frac{1}{n}
+ \rightarrow
+ \infty,
+ \end{equation*}
+ und somit ist $\Re(s) > 1$.
+ \end{frame}
+
+ \section{Analytische Fortsetzung}
+ \begin{frame}
+ \frametitle{Plan für die Analytische Fortsetzung von $\zeta(s)$}
+ \begin{center}
+ \input{../continuation_overview.tikz.tex}
+ \end{center}
+ \end{frame}
+ \begin{frame}
+ \frametitle{Fortsetzung auf $\Re(s) > 0$}
+ Dirichletsche Etafunktion ist
+ \begin{equation*}\label{zeta:equation:eta}
+ \eta(s)
+ =
+ \sum_{n=1}^{\infty}
+ \frac{(-1)^{n-1}}{n^s},
+ \end{equation*}
+ und konvergiert im Bereich $\Re(s) > 0$.
+ \end{frame}
+ \begin{frame}
+ \frametitle{Fortsetzung auf $\Re(s) > 0$}
+ \begin{align}
+ \zeta(s)
+ &=
+ \RD{
+ \sum_{n=1}^{\infty}
+ \frac{1}{n^s} \label{zeta:align1}
+ }
+ \\
+ \frac{1}{2^{s-1}}
+ \zeta(s)
+ &=
+ \BL{
+ \sum_{n=1}^{\infty}
+ \frac{2}{(2n)^s} \label{zeta:align2}
+ }
+ \end{align}
+ \pause
+ \eqref{zeta:align1} - \eqref{zeta:align2}:
+ \begin{align*}
+ \left(1 - \frac{1}{2^{s-1}} \right)
+ \zeta(s)
+ &=
+ \RD{\frac{1}{1^s}}
+ \underbrace{-\BL{\frac{2}{2^s}} + \RD{\frac{1}{2^s}}}_{-\frac{1}{2^s}}
+ + \RD{\frac{1}{3^s}}
+ \underbrace{-\BL{\frac{2}{4^s}} + \RD{\frac{1}{4^s}}}_{-\frac{1}{4^s}}
+ \ldots
+ \\
+ &= \eta(s)
+ \end{align*}
+ \end{frame}
+ \begin{frame}
+ \frametitle{Fortsetzung auf $\Re(s) > 0$}
+ Somit haben wir die Fortsetzung gefunden als
+ \begin{equation} \label{zeta:equation:fortsetzung1}
+ \zeta(s)
+ :=
+ \left(1 - \frac{1}{2^{s-1}} \right)^{-1} \eta(s).
+ \end{equation}
+ \end{frame}
+ \begin{frame}
+ \frametitle{Spiegelungseigenschaft für $\Re(s) < 0$}
+ \begin{equation*}\label{zeta:equation:functional}
+ \frac{\Gamma \left( \frac{s}{2} \right)}{\pi^{\frac{s}{2}}}
+ \zeta(s)
+ =
+ \frac{\Gamma \left( \frac{1-s}{2} \right)}{\pi^{\frac{1-s}{2}}}
+ \zeta(1-s).
+ \end{equation*}
+ \end{frame}
+ %TODO maybe explain gamma-fct
+
+ \section{Euler Produkt und Primzahlen}
+ \begin{frame}
+ \frametitle{Wieso ist die Zeta Funktion so bekannt?}
+ \begin{itemize}
+ \item Interessante Funktionswerte z.B. $\zeta(2) = \frac{\pi^2}{6}$
+ \item Primzahlenverteilung (Riemannhypothese)
+ \item Forschungsgebiet der analytischen Zahlentheorie seit dem 18. Jahrhundert
+ \item ...
+ \end{itemize}
+ \end{frame}
+ \begin{frame}
+ \frametitle{Euler Produkt: Verbindung von Zeta und Primzahlen}
+ \begin{equation*}
+ \zeta(s)
+ =
+ \sum_{n=1}^\infty
+ \frac{1}{n^s}
+ =
+ \prod_{p \in P}
+ \frac{1}{1-p^{-s}}
+ \end{equation*}
+ \pause
+ Geometrische Reihe
+ \begin{equation*}
+ \prod_{p \in P}
+ \frac{1}{1-p^{-s}}
+ =
+ \prod_{p \in P}
+ \left(
+ 1
+ +
+ \frac{1}{p^s}
+ +
+ \frac{1}{p^{2s}}
+ +
+ \frac{1}{p^{3s}}
+ +
+ \ldots
+ \right)
+ \end{equation*}
+ \pause
+ Erste Terme ausmultiplizieren
+ \begin{align*}
+ \left(
+ 1
+ +
+ \RD{\frac{1}{2^s}}
+ +
+ \GN{\frac{1}{2^{2s}}}
+ +
+ \frac{1}{2^{3s}}
+ +
+ \ldots
+ \right)
+ \left(
+ 1
+ +
+ \BL{\frac{1}{3^s}}
+ +
+ \frac{1}{3^{2s}}
+ +
+ \frac{1}{3^{3s}}
+ +
+ \ldots
+ \right)
+ \left(
+ 1
+ +
+ \YE{\frac{1}{5^s}}
+ +
+ \frac{1}{5^{2s}}
+ +
+ \frac{1}{5^{3s}}
+ +
+ \ldots
+ \right)
+ \ldots
+ \\
+ =
+ 1
+ +
+ \RD{\frac{1}{2^s}}
+ +
+ \BL{\frac{1}{3^s}}
+ +
+ \GN{\frac{1}{4^s}}
+ +
+ \YE{\frac{1}{5^s}}
+ +
+ \ldots
+ \end{align*}
+ \end{frame}
+ \begin{frame}
+ \frametitle{Primzahlfunktion}
+ \begin{center}
+ \scalebox{0.5}{\input{../primzahlfunktion.pgf}}
+ \end{center}
+ \end{frame}
+
+
+ \section{Darstellungen}
+
+ \begin{frame}
+ \frametitle{Farbcodierung}
+ \begin{center}
+ \scalebox{0.6}{\input{zeta_color_plot.pgf}}
+ \end{center}
+ \end{frame}
+
+ \begin{frame}
+ \frametitle{Konstanter Realteil $\Re(s)=-1$ und $\Im(s)=0\ldots40$}
+ \begin{center}
+ \scalebox{0.6}{\input{../zeta_re_-1_plot.pgf}}
+ \end{center}
+ \end{frame}
+ \begin{frame}
+ \frametitle{Konstanter Realteil $\Re(s)=0$ und $\Im(s)=0\ldots40$}
+ \begin{center}
+ \scalebox{0.6}{\input{../zeta_re_0_plot.pgf}}
+ \end{center}
+ \end{frame}
+ \begin{frame}
+ \frametitle{Konstanter Realteil $\Re(s)=0.5$ und $\Im(s)=0\ldots40$}
+ \begin{center}
+ \scalebox{0.6}{\input{../zeta_re_0.5_plot.pgf}}
+ \end{center}
+ \end{frame}
+
+\end{document}
+
diff --git a/buch/papers/zeta/presentation/youtube_screenshot.png b/buch/papers/zeta/presentation/youtube_screenshot.png
new file mode 100644
index 0000000..434041b
--- /dev/null
+++ b/buch/papers/zeta/presentation/youtube_screenshot.png
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diff --git a/buch/papers/zeta/presentation/zeta_color_plot-img0.png b/buch/papers/zeta/presentation/zeta_color_plot-img0.png
new file mode 100644
index 0000000..b8c7298
--- /dev/null
+++ b/buch/papers/zeta/presentation/zeta_color_plot-img0.png
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diff --git a/buch/papers/zeta/presentation/zeta_color_plot.pgf b/buch/papers/zeta/presentation/zeta_color_plot.pgf
new file mode 100644
index 0000000..0fd7cb8
--- /dev/null
+++ b/buch/papers/zeta/presentation/zeta_color_plot.pgf
@@ -0,0 +1,402 @@
+%% Creator: Matplotlib, PGF backend
+%%
+%% To include the figure in your LaTeX document, write
+%% \input{<filename>.pgf}
+%%
+%% Make sure the required packages are loaded in your preamble
+%% \usepackage{pgf}
+%%
+%% and, on pdftex
+%% \usepackage[utf8]{inputenc}\DeclareUnicodeCharacter{2212}{-}
+%%
+%% or, on luatex and xetex
+%% \usepackage{unicode-math}
+%%
+%% Figures using additional raster images can only be included by \input if
+%% they are in the same directory as the main LaTeX file. For loading figures
+%% from other directories you can use the `import` package
+%% \usepackage{import}
+%%
+%% and then include the figures with
+%% \import{<path to file>}{<filename>.pgf}
+%%
+%% Matplotlib used the following preamble
+%%
+\begingroup%
+\makeatletter%
+\begin{pgfpicture}%
+\pgfpathrectangle{\pgfpointorigin}{\pgfqpoint{6.400000in}{4.800000in}}%
+\pgfusepath{use as bounding box, clip}%
+\begin{pgfscope}%
+\pgfsetbuttcap%
+\pgfsetmiterjoin%
+\definecolor{currentfill}{rgb}{1.000000,1.000000,1.000000}%
+\pgfsetfillcolor{currentfill}%
+\pgfsetlinewidth{0.000000pt}%
+\definecolor{currentstroke}{rgb}{1.000000,1.000000,1.000000}%
+\pgfsetstrokecolor{currentstroke}%
+\pgfsetdash{}{0pt}%
+\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}%
+\pgfpathlineto{\pgfqpoint{6.400000in}{0.000000in}}%
+\pgfpathlineto{\pgfqpoint{6.400000in}{4.800000in}}%
+\pgfpathlineto{\pgfqpoint{0.000000in}{4.800000in}}%
+\pgfpathclose%
+\pgfusepath{fill}%
+\end{pgfscope}%
+\begin{pgfscope}%
+\pgfsetbuttcap%
+\pgfsetmiterjoin%
+\definecolor{currentfill}{rgb}{1.000000,1.000000,1.000000}%
+\pgfsetfillcolor{currentfill}%
+\pgfsetlinewidth{0.000000pt}%
+\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}%
+\pgfsetstrokecolor{currentstroke}%
+\pgfsetstrokeopacity{0.000000}%
+\pgfsetdash{}{0pt}%
+\pgfpathmoveto{\pgfqpoint{2.588156in}{0.528000in}}%
+\pgfpathlineto{\pgfqpoint{3.971844in}{0.528000in}}%
+\pgfpathlineto{\pgfqpoint{3.971844in}{4.224000in}}%
+\pgfpathlineto{\pgfqpoint{2.588156in}{4.224000in}}%
+\pgfpathclose%
+\pgfusepath{fill}%
+\end{pgfscope}%
+\begin{pgfscope}%
+\pgfpathrectangle{\pgfqpoint{2.588156in}{0.528000in}}{\pgfqpoint{1.383688in}{3.696000in}}%
+\pgfusepath{clip}%
+\pgfsys@transformshift{2.588156in}{0.528000in}%
+\pgftext[left,bottom]{\includegraphics[interpolate=true,width=1.390000in,height=3.700000in]{zeta_color_plot-img0.png}}%
+\end{pgfscope}%
+\begin{pgfscope}%
+\pgfsetbuttcap%
+\pgfsetroundjoin%
+\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}%
+\pgfsetfillcolor{currentfill}%
+\pgfsetlinewidth{0.803000pt}%
+\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}%
+\pgfsetstrokecolor{currentstroke}%
+\pgfsetdash{}{0pt}%
+\pgfsys@defobject{currentmarker}{\pgfqpoint{0.000000in}{-0.048611in}}{\pgfqpoint{0.000000in}{0.000000in}}{%
+\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}%
+\pgfpathlineto{\pgfqpoint{0.000000in}{-0.048611in}}%
+\pgfusepath{stroke,fill}%
+}%
+\begin{pgfscope}%
+\pgfsys@transformshift{2.588156in}{0.528000in}%
+\pgfsys@useobject{currentmarker}{}%
+\end{pgfscope}%
+\end{pgfscope}%
+\begin{pgfscope}%
+\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}%
+\pgfsetstrokecolor{textcolor}%
+\pgfsetfillcolor{textcolor}%
+\pgftext[x=2.588156in,y=0.430778in,,top]{\color{textcolor}\rmfamily\fontsize{8.000000}{9.600000}\selectfont \(\displaystyle {-10}\)}%
+\end{pgfscope}%
+\begin{pgfscope}%
+\pgfsetbuttcap%
+\pgfsetroundjoin%
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+\begin{pgfscope}%
+\pgfsetbuttcap%
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diff --git a/buch/papers/zeta/python/plot_zeta.py b/buch/papers/zeta/python/plot_zeta.py
new file mode 100644
index 0000000..53097c5
--- /dev/null
+++ b/buch/papers/zeta/python/plot_zeta.py
@@ -0,0 +1,39 @@
+import numpy as np
+from mpmath import zeta
+import matplotlib.pyplot as plt
+from matplotlib import colors
+import matplotlib
+matplotlib.use("pgf")
+matplotlib.rcParams.update(
+ {
+ "pgf.texsystem": "pdflatex",
+ "font.family": "serif",
+ "font.size": 8,
+ "text.usetex": True,
+ "pgf.rcfonts": False,
+ "axes.unicode_minus": False,
+ }
+)
+
+print(zeta(-1))
+print(zeta(-1 + 2j))
+
+re_values = np.arange(-10, 5, 0.04)
+im_values = np.arange(-20, 20, 0.04)
+plot_matrix = np.zeros((len(im_values), len(re_values), 3))
+for im_i, im in enumerate(im_values):
+ print(im_i)
+ for re_i, re in enumerate(re_values):
+ z = complex(zeta(re + 1j*im))
+ h = (np.angle(z) + np.pi) / (2*np.pi)
+ v = np.abs(z)
+ s = 1.0
+ plot_matrix[im_i, re_i] = [h, s, v]
+
+log10_v = np.log10(plot_matrix[:, :, 2])
+log10_v += np.abs(np.min(log10_v))
+plot_matrix[:, :, 2] = (log10_v) / np.max(log10_v)
+plt.imshow(colors.hsv_to_rgb(plot_matrix), extent=[re_values.min(), re_values.max(), im_values.min(), im_values.max()])
+plt.xlabel("$\Re$")
+plt.ylabel("$\Im$")
+plt.savefig(f"zeta_color_plot.pgf")
diff --git a/buch/papers/zeta/python/plot_zeta2.py b/buch/papers/zeta/python/plot_zeta2.py
new file mode 100644
index 0000000..b730703
--- /dev/null
+++ b/buch/papers/zeta/python/plot_zeta2.py
@@ -0,0 +1,31 @@
+import numpy as np
+from mpmath import zeta
+import matplotlib.pyplot as plt
+import matplotlib
+matplotlib.use("pgf")
+matplotlib.rcParams.update(
+ {
+ "pgf.texsystem": "pdflatex",
+ "font.family": "serif",
+ "font.size": 8,
+ "text.usetex": True,
+ "pgf.rcfonts": False,
+ "axes.unicode_minus": False,
+ }
+)
+# const re plot
+re_values = [-1, 0, 0.5]
+im_values = np.arange(0, 40, 0.04)
+buf = np.zeros((len(re_values), len(im_values), 2))
+for im_i, im in enumerate(im_values):
+ print(im_i)
+ for re_i, re in enumerate(re_values):
+ z = complex(zeta(re + 1j*im))
+ buf[re_i, im_i] = [np.real(z), np.imag(z)]
+
+for i in range(len(re_values)):
+ plt.figure()
+ plt.plot(buf[i,:,0], buf[i,:,1], label=f"$\Re={re_values[i]}$")
+ plt.xlabel("$\Re$")
+ plt.ylabel("$\Im$")
+ plt.savefig(f"zeta_re_{re_values[i]}_plot.pgf")
diff --git a/buch/papers/zeta/python/primzahlfunktion.py b/buch/papers/zeta/python/primzahlfunktion.py
new file mode 100644
index 0000000..9434de9
--- /dev/null
+++ b/buch/papers/zeta/python/primzahlfunktion.py
@@ -0,0 +1,24 @@
+import matplotlib.pyplot as plt
+import numpy as np
+
+primzahlfunktion = [0, 0, 0, 0]
+x = [0, 1-1e-12, 1, 2-1e-12]
+x_last = 1
+value = 0
+for i in range(2, 30, 1):
+ new_value = value + 1
+ for j in range(2, i, 1):
+ if i % j == 0:
+ new_value = value
+ value = new_value
+ primzahlfunktion.append(new_value)
+ x_last += 1
+ x.append(x_last)
+ primzahlfunktion.append(new_value)
+ x.append(x_last + 1 - 1e-12)
+
+
+plt.rcParams.update({"pgf.texsystem": "pdflatex"})
+plt.plot(x, primzahlfunktion)
+plt.show()
+
diff --git a/buch/papers/zeta/zeta_re_-1_plot.pgf b/buch/papers/zeta/zeta_re_-1_plot.pgf
new file mode 100644
index 0000000..dd15ba1
--- /dev/null
+++ b/buch/papers/zeta/zeta_re_-1_plot.pgf
@@ -0,0 +1,1147 @@
+%% Creator: Matplotlib, PGF backend
+%%
+%% To include the figure in your LaTeX document, write
+%% \input{<filename>.pgf}
+%%
+%% Make sure the required packages are loaded in your preamble
+%% \usepackage{pgf}
+%%
+%% and, on pdftex
+%% \usepackage[utf8]{inputenc}\DeclareUnicodeCharacter{2212}{-}
+%%
+%% or, on luatex and xetex
+%% \usepackage{unicode-math}
+%%
+%% Figures using additional raster images can only be included by \input if
+%% they are in the same directory as the main LaTeX file. For loading figures
+%% from other directories you can use the `import` package
+%% \usepackage{import}
+%%
+%% and then include the figures with
+%% \import{<path to file>}{<filename>.pgf}
+%%
+%% Matplotlib used the following preamble
+%%
+\begingroup%
+\makeatletter%
+\begin{pgfpicture}%
+\pgfpathrectangle{\pgfpointorigin}{\pgfqpoint{6.400000in}{4.800000in}}%
+\pgfusepath{use as bounding box, clip}%
+\begin{pgfscope}%
+\pgfsetbuttcap%
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new file mode 100644
index 0000000..3ac7df8
--- /dev/null
+++ b/buch/papers/zeta/zeta_re_0.5_plot.pgf
@@ -0,0 +1,1206 @@
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+%%
+%% and, on pdftex
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